PREDATOR-PREY MODEL WITH CONSTANT RATE OF HARVESTING
NURUL AINA BINTI ARIFIN
A thesis submitted in fulfillment of the
requirements for the awards of the degree of
Master of Science (Mathematics)
Faculty of Science
Universiti Teknologi Malaysia
JANUARY 2014
iii
Special dedicate to my beloved family
Arifin Bin Abdullah
Rohani Binti Dali
Mohd Fakharuddin Bin Arifin
Nurulain Hanis Binti Shahruni
Mohamad Amirudin Bin Arifin
and
Nurul Atiqah Arifin
Thanks for all the supports, sacrifices, patients and willingness to share my dreams
with me...
To my honoured supervisor, Dr Faridah Mustapha, my friends (Mira, Hidayah,
Yana, Azira, Amiza and Aisya) and all UTM lecturers
Thank you for everything you taught me. Giving me inspiration to be knowledge
person.
iv
ACKNOWLEDGMENT
All praise belongs to Allah (SWT), The Lord of the Universe, without the
health, strength and perseverance he gave, I would not be able to complete this
thesis.
First and foremost, I would like to express my gratefulness and appreciation
to my supervisor Dr. Faridah Mustapha for her guidance and support that she had
given me along the completion of this project.
I also feel grateful to PSZ for providing me the information for my research
study.
Finally, I also would like to thank all my beloved persons in my life that
inspire and motivate me a lot throughout all the hard works while carrying out this
research, especially to my beloved parents, Arifin Abdullah and Rohani Dali, my
brother, Fakharudin, my little brother and sister, Amirudin and Atiqah. Last but not
least, thanks to all my friends for sharing ideas and giving encouragement.
v
ABSTRACT
Predator-prey model is the first model to illustrate the interaction between
predators and prey. It is a topic of great interest for many ecologists and
mathematicians. This model assumes that the predator populations have negative
effects on the prey populations. The generalized equation of this model is the
response of the populations would be proportional to the product of their population
densities. Prey population grows with limited by carrying capacity, K and it is called
the logistic equation. Thus, in this research, there are four different cases are
analyzed which are predator-prey model, predator-prey model with harvesting in
prey, predator-prey model with harvesting in predator and predator-prey model with
harvesting in both populations. Systems of ordinary differential equation are used for
all models. The objectives of this research are i) to study the concept of Lotka-
Volterra predator-prey model, ii) to analyze the predator-prey model with constant
rate of harvesting in prey, iii) to analyze the predator-prey model with constant rate
of harvesting in predator, iv) to analyze the predator-prey model with constant rate of
harvesting in both populations. In analyzing all four models, equilibrium points will
be obtained and analyzed for the stability by using Routh-Hurwitz Criteria. Lastly,
some numerical examples and graphical analysis are shown to illustrate the stability
of the stable equilibrium points and the effects of harvesting to the systems.
vi
ABSTRAK
Model Pemangsa-Mangsa merupakan model yang pertama untuk
menggambarkan interaksi antara pemangsa dan mangsa. Ia adalah satu topik yang
menarik minat besar bagi semua ahli ekologi dan ahli matematik. Model ini
mengandaikan populasi pemangsa memberi kesan negatif kepada populasi mangsa.
Persamaan umum model ini adalah tindak balas sesuatu populasi yang berkadar
dengan hasil kepadatan sesuatu populasi. Populasi mangsa membesar dengan
dihadkan oleh keupayaan membawa, K dan ia dipanggil sebagai persamaan logistik.
Oleh itu, dalam kajian ini, terdapat empat kes berbeza yang dianalisis iaitu model
pemangsa-mangsa, model pemangsa-mangsa dengan penangkapan dalam mangsa,
model pemangsa-mangsa dengan penangkapan dalam pemangsa dan model
pemangsa-mangsa dengan penangkapan dalam kedua-dua populasi. Sistem
persamaan pembezaan biasa digunakan untuk keseluruhan model. Objektif kajian ini
adalah i) untuk mengkaji konsep model Lotka-Volterra Pemangsa-mangsa, ii) untuk
menganalisis pemangsa-mangsa model dengan kadar tetap penangkapan dalam
mangsa, iii) untuk menganalisis pemangsa-mangsa model dengan kadar tetap
penangkapan dalam pemangsa, iv) untuk menganalisis pemangsa-mangsa model
dengan kadar tetap penangkapan dalam kedua-dua populasi. Dalam menganalisis
keempat-empat model tersebut, titik keseimbangan akan diperolehi dan
kestabilannya akan dianalisis dengan menggunakan Kriteria Routh-Hurwitz. Akhir
sekali, beberapa contoh berangka dan analisis graf ditunjukkan untuk
memperlihatkan kestabilan titik keseimbangan dan kesan penangkapan dalam
sistem.
vii
TABLE OF CONTENTS
CHAPTER TITLE PAGE
DECLARATION
DEDICATION
ACKNOWLEDGEMENT
ABSTRACT
ABSTRAK
TABLE OF CONTENTS
LIST OF SYMBOLS
LIST OF FIGURES
LIST OF TABLES
LIST OF APPENDICES
ii
iii
iv
v
vi
vii
xi
xii
xv
xvi
1 INTRODUCTION
1.1 Background of Research
1.2 Problem Statement
1.3 Objectives of Research
1.4 Scope of Research
1.5 Significant of Research
1
3
3
4
4
2 LITERATURE REVIEW
2.1 Introduction
2.2 Mathematical Modeling
5
6
viii
2.3 Predator-Prey Interaction
2.4 Lotka-Volterra Predator-Prey Model
2.5 Predator-Prey Model with Constant Rate of
Harvesting
2.5.1 Definition of Harvesting in Population
Dynamics
2.5.2 Stability with Harvesting
7
8
9
10
11
3 RESEARCH METHODOLOGY
3.1 Introduction
3.2 Nullclines
3.3 Equilibrium Analysis
3.4 Eigenvalues
3.4.1 Asymptotically Stability
3.5 Routh – Hurwitz Criteria
3.6 Numerical Method to Solve Ordinary Differential
Equation
14
15
16
17
19
20
21
4 PREDATOR-PREY MODEL
4.1 Introduction
4.2 The Model
4.3 Equilibrium Points of Predator-Prey Model
4.4 Analyzing the Model
4.5 Numerical Examples
23
24
26
28
33
ix
5 PREDATOR-PREY MODEL WITH CONSTANT
RATE OF PREY HARVESTING
5.1 Introduction
5.2 The Model
5.3 Equilibrium Points of Predator-Prey Model with
Harvesting in Prey
5.4 Analyzing the Model
5.5 Numerical Examples
38
39
42
45
51
6 PREDATOR-PREY MODEL WITH CONSTANT
RATE OF PREDATOR HARVESTING
6.1 Introduction
6.2 The Model
6.3 Equilibrium Points of Predator-Prey Model with
Harvesting in Predator
6.4 Analyzing the Model
6.5 Numerical Examples
58
59
62
65
69
7 PREDATOR-PREY MODEL WITH CONSTANT
RATE OF BOTH POPULATION HARVESTING
7.1 Introduction
7.2 The Model
7.3 Equilibrium Points of Predator-Prey Model with
Harvesting in Both Populations
7.4 Analyzing the Model
7.5 Numerical Examples
73
74
77
79
83
x
8 CONCLUSION AND RECOMMENDATIONS
8.1 Introduction
8.2 Research Conclusion
8.3 Recommendations
87
88
90
REFERENCES 91
APPENDICES 94
xi
LIST OF SYMBOLS
( )x t - Prey population
( )y t - Predator population
xH - Prey harvesting
yH - Predator harvesting
1,2 - Eigenvalues of equilibrium point
r - Growth rate of prey
K - Carrying capacity of the prey.
- Rate of consumption of prey by predator.
- Conversion of prey consumed into predator reproduction rate.
c - Death rate of predator
xii
LIST OF FIGURES
FIGURE
NO. TITLE PAGE
Figure 3.1 Phase portraits for real or distinct eigenvalues 18
Figure 3.2 Phase portraits for complex eigenvalues 19
Figure 4.1 The death rate of predator, K versus the conversion of prey
into predator reproduction rate, c with fixed values for 1r
and 1 .
32
Figure 4.2 Phase portraits for a nonlinear system (4.10) 34
Figure 4.3 The graph show population densities of prey and predator
versus time, 1, 1, 2, 0.05 and 0.3r K c
35
Figure 4.4 Phase portraits for a nonlinear system (4.11) 36
Figure 4.5 The graph show population densities of prey and predator
versus time, 1, 1, 10, 0.05 and 0.3r K c
37
Figure 5.1 Constant rate of prey harvesting, xH versus growth rate of prey
and carrying capacity, rK
43
Figure 5.2 Constant rate of prey harvesting, xH versus death rate of
predator, c with 1, 2, 1 and 0.05r K .
50
Figure 5.3 Phase portraits for a nonlinear system (5.23) 52
xiii
Figure 5.4 Phase portraits for a nonlinear system with 0.07c and
0.45xH
53
Figure 5.5 The graphs show the population densities of prey and predator
versus time, 1, 2, 1, 0.05r K . a) 0.09, 0.4xc H
b) 0.07, 0.45xc H
53
Figure 5.6 Phase portraits for a nonlinear system (5.24) 55
Figure 5.7 Phase portraits for a nonlinear system with 0.07 andc
0.3xH
56
Figure 5.8 The graphs show the population densities of prey and predator
versus time, 1, 2, 1, 0.05r K . a) 0.08, 0.2xc H
b) 0.07, 0.3xc H
56
Figure 6.1 Constant rate of predator harvesting, yH versus growth rate of
prey, r with 10, 1, 0.05 and 0.3K c
64
Figure 6.2 Phase portraits of nonlinear system (6.20) with 1 andr
0.01yH
70
Figure 6.3 Phase portraits with 2 and 0.02yr H 71
Figure 6.4 The graph show the population densities of prey and predator
versus time 10, 1, 0.05, 0.3K c
a) 1 and 0.01yr H b) 2 and 0.02yr H
71
Figure 7.1 Constant rate of prey harvesting, xH versus growth rate of prey
and carrying capacity, rK
1, 0.05, 0.3 and 0.02yc H
78
xiv
Figure 7.2 Constant rate of prey harvesting, xH versus the conversion of
prey consumed into predator reproduction rate, with
( 1, 20, 1, 0.3 and 0.02yr K c H ).
85
Figure 7.3 The graph show the population densities of prey and predator
versus time, 1, 100, 1, 0.05, 0.3, 0.01 xr K c H
and 0.02yH
86
xv
LIST OF TABLES
TABLE
NO. TITLE PAGE
5.1 Parameters used in predator-prey model with constant
rate of harvesting in prey
41
8.1 Summarization on the stability of equilibrium points with
the conditions for predator-prey model without
harvesting, predator-prey model with harvesting in prey,
predator-prey model with harvesting in both populations.
89
xvi
LIST OF APPENDICES
APPENDIX TITLE PAGE
A Calculation of Eigenvalues for Each Equilibrium Points
and Calculation of Fourth Order Runge Kutta Using Maple
Software.
94
B Phase Portraits of the Predator-Prey Model with Constant
Rate of Prey Harvesting using Matlab Software.
96
CHAPTER 1
INTRODUCTION
1.1 Background of Research
In the ecology system, the predator-prey model is amongst the oldest studies
and also the first model to illustrate the interaction between predators and prey. This
model assume that the predator populations have negative effects on the prey
populations and this system was formulated by Vito Volterra, an Italian
mathematician and Alfred Lotka, an American mathematical biologist in 1925
(Boyce and DiPrima, 2010).
There are several known predator-prey models such as Lotka-Volterra model,
Logistic equation, Holling Tanner Type 2 and Type 3 models, Yodzis model and so
on. This study only focuses on the most well known Lotka-Volterra model. This
model has been analyzed by various text books in dynamical systems, mathematical
biology, ecology, differential equations etc. The generalized equation of this model is
the response of the populations would be proportional to the product of their
population densities. The prey population grows infinitely in the absence of
predators. Therefore, the logistic equation or often called as carrying capacity of the
environment was added to the prey equation given,
2
( )( ) 1 ( ) ( ),
( ) ( ) ( ),
dx x trx t x t y t
dt K
dyx t y t cy t
dt
(1.1)
where
( )x t denotes prey population.
( )y t denotes predator population.
r denotes the growth rate of prey.
K denotes carrying capacity.
denotes the rate of consumption of prey by predator.
denotes the conversion of prey consumed into predator reproduction
rate.
c denotes death rate of the predator.
Our study also continues by adding constant rate of harvesting to the model.
In the previous research, Brauer (1977), Dai and Tang (1998), Martin and Ruan
(2001), Kar and Pahari (2006), Syamsuddin and Malik (2008), Xia et al. (2009) and
Agarwal and Pathak (2012) had analyzed the constant rate of harvesting either prey
or predator population. According to Kar and Pahari (2006), many of interesting
dynamical behaviors such as the stability of the equilibrium points, existence of Hopf
bifurcation and limit cycles have been observed. Therefore, this study concern on
analyze the stability of the model by adding the harvesting to both populations by
using the method of differential equation.
Lotka-Volterra predator-prey model plays a crucial role in the population
dynamics and also one of the major advanced theories introduced by Lotka and
Volterra. They assumed the response of the population would be proportional to the
product of their population densities without any delayed and constant rate of
harvesting. The model formulation is in the equation (1.1). Some of the studies found
that the rate of harvesting has been used to control the increasing of population and
as a controller of the population density (Ouncharoen et al., 2010). This means that
3
harvesting can make whether the population increases or decreases for a continuity
yields but it will limited by a carrying capacity in equation (1.1).
In this project, the constant rate of harvesting will be added to either preys,
predators or both populations and we will analyze the stability of the equilibrium
points of the predator-prey model.
1.2 Problem Statement
This study is focused on how constant rate of harvesting affect the dynamics
of the predator-prey system.
1.3 Objectives of Research
The objectives of this research are:
1. To study the concept of Lotka-Volterra predator-prey model.
2. To analyze the predator-prey model with constant rate of harvesting in
prey.
3. To analyze the predator-prey model with constant rate of harvesting in
predator.
4. To analyze the predator-prey model with constant rate of harvesting in
both populations.
4
1.4 Scope of Research
The main scope of this research is to analyze the predator-prey model by with
the constant rate of harvesting. In this research, we shall only focus on two
populations which are prey and predator. We will formulate the model by adding the
constant rate of harvesting prey, predator or both population and also find how this
factor affects to the stability of equilibrium points in predator-prey model.
1.5 Significant of Research
The findings of this research is useful for the mathematicians who are
interested in the ecology fields because this research will give us more
understanding in predator-prey model with or without harvesting and the effects on
the stability of the population. The result obtained can be a guide by applying to
another predation model in population field.
91
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“Harvesting a Prey Population”,
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